Floquet Topological Insulator in Semiconductor Quantum Wells

Netanel H. Lindner1,2, Gil Refael1,2, Victor Galitski3,4

1) Institute of Quantum Information, California Institute of Technology, Pasadena, CA 91125, USA.2) Department of Physics, California Institute of Technology, Pasadena, CA 91125, USA.

3) Condensed Matter Theory Center, Department of Physics,University of Maryland, College Park, Maryland 20742, USA. and

4) Joint Quantum Institute, Department of Physics,University of Maryland, College Park, Maryland 20742, USA

Topological phase transitions between a conventional insulator and a state of matter with topolog-ical properties have been proposed and observed in mercury telluride - cadmium telluride quantumwells. We show that a topological state can be induced in such a device, initially in the trivialphase, by irradiation with microwave frequencies, without closing the gap and crossing the phasetransition. We show that the quasi-energy spectrum exhibits a single pair of helical edge states. Thevelocity of the edge states can be tuned by adjusting the intensity of the microwave radiation. Wediscuss the necessary experimental parameters for our proposal. This proposal provides an exampleand a proof of principle of a new non-equilibrium topological state, Floquet topological insulator,introduced in this paper.

Topological phases of matter have captured our imag-ination over the past few years, with tantalizing proper-ties such as robust edge modes and exotic non-Abelianexcitations [1, 2], and potential applications ranging fromsemiconductor spintronics [3] to topological quantumcomputation [4]. The discovery of topological insula-tors in solid-state devices such as HgTe/CdTe quantumwells [5, 6], and in materials such as Bi2Te3, Bi2Sn3 [7–9] brings us closer to employing the unique properties oftopological phases in technological applications.

Despite this success, however, the choice of materi-als that exhibit these unique topological properties re-mains rather scarce. In most cases we have to rely onserendipity in looking for topological materials in solid-state structures and our means to engineer Hamiltoniansthere are very limited. Therefore, to develop new meth-ods to achieve and control topological structures at willwould be of great importance.

Our work demonstrates that such new methods areindeed possible in non-equilibrium, where external time-dependent perturbations represent a rich and versatile re-source that can be utilized to achieve topological spectrain systems that are topologically trivial in equilibrium. Inparticular, we show that periodic-in-time perturbationsmay give rise to new differential operators with topolog-ical insulator spectra, dubbed Floquet topological insu-lators (FTI), that exhibit chiral edge currents in non-equilibrium and possess other hallmark phenomena asso-ciated with topological phases. These ideas, put togetherwith the highly developed technology for controlling low-frequency electromagnetic modes, can enable devices inwhich fast switching of edge state transport is possible.Moreover, the spectral properties of the edge states, i.e.,their velocity, and the bandgap of the bulk insulator, canbe easily controlled. On a less applied perspective, thefast formation of the Floquet topological insulators in re-sponse to the external field opens a path to study quench

dynamics of topological states in solid-state devices.The Floquet topological insulators discussed here

share many features discussed in some previous works:The idea of achieving topological states in a periodicHamiltonian was also explored from the perspective ofquantum walks in Ref. 10. Also, a similar philosophy ledto proposals for the realization of topological phases incold-atom systems: a quantum Hall state using a stro-boscopic quadrupole field [11] and a topological insulatorstate using a Ramann-scattering induced spin-orbit cou-pling [12]. Also, Ref. [13] proposed to use a circularly-polarized light field to induce a Hall current in graphene.Another useful analogy for our work is the formation ofzero-resistance state in Hall bars at low magnetic fieldsusing RF radiation [14–17]. In our case, it is not theresistance of the bulk that vanishes, but rather the re-sistance of the edges, which becomes finite and universalupon application of the light field. As we were finaliz-ing our draft, we also became aware of Ref. 18, whichproposed to use a periodic modulation in the form of acircularly polarized light to change the Chern number inthe Haldane model [19].

DEFINITION OF A FLOQUET TOPOLOGICALINSULATOR

Let us first provide a general construction and def-inition for a Floquet topological insulator in a genericlattice model, and then discuss a specific realization:a HgTe/CdTe quantum well. The generic many-bodyHamiltonian of interest is

H(t) =∑k∈BZ

Hnm(k, t)c†n,kcm,k + h. c., (1)

where c†n,k and cm,k are fermion creation/annihilationoperators, k is the momentum defined in the Brillouin

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zone, and the Latin indices, n,m = 1, 2, . . . N label someinternal degrees of freedom (e.g., spin, sublattice, layerindices, etc.). The N × N k-dependent matrix H(k, t)is determined by lattice hoppings and/or external fields,which are periodic in time, H(T + t) = H(t).

First, we recall that without the time-dependence, thetopological classification reduces to an analysis of the ma-trix function, H(k), and is determined by its spectrum.[20, 21]. An interesting question is whether a topologi-cal classification is possible in non-equilibrium, i.e., whenthe single-particle Hamiltonian, H(k, t), in Eq. (1) doeshave an explicit time-dependence and whether there areobservable physical phenomena associated with this non-trivial topology. Consider the single-particle Schrodingerequation associated with Eq. (1):[H(k, t)− iI∂t

]Ψk(t) = 0, with H(k, t) = H(k, t+ T )

(2)The Bloch-Floquet theory states that the solutions toEq. (2) have the form Ψk(t) = Sk(t)Ψk(0), where theunitary evolution is given by the product of a periodicunitary part and a Floquet exponential

Sk(t) = Pk(t) exp[−iHF (k)t

], with Pk(t) = Pk(t+ T )

(3)where HF (k) is a self-adjoint time-independent matrix,associated with the Floquet operator

[H(k, t)− iI∂t

]acting in the space of periodic functions Φ(t) = Φ(t+T ),where it leads to a time-independent eigenvalue prob-lem,

[H(k, t)− iI∂t

]Φ(k, t) = ε(k)Φ(k, t). The quasi-

energies ε(k) are the eigenvalues of the matrix HF (k) inEq. (3), and in the cases of interest can be divided intoseparate bands. The full single-particle wave-function istherefore given by Ψ(t) = e−iεtΦ(t). Note that the quasi-energies are defined modulo the frequency ω = 2π/T .

The Floquet topological insulator is defined throughthe topological properties of the time-independent Flo-quet operator HF (k), in accordance with the exist-ing topological classification of equilibrium Hamiltonians[20, 21]. Most importantly, we show below that the FTIis not only a mathematical concept, but it has immediatephysical consequences in the form of robust edge statesthat appear in a finite system in the non-equilibriumregime. The density profile of these hallmark bound-ary modes was found to be only weakly time-dependent[through the envelope function determined by the matrixPk(t) in Eq. 3]. Furthermore, we explicitly demonstratewithin a specific model that not only topological insulatorproperties may survive in non-equilibrium, but they canbe induced via a simple non-equilibrium perturbation inan otherwise topologically trivial system. Using the spe-cific model describing a HgTe/CdTe quantum wells, weshow that a carefully chosen non-equilibrium perturba-tion may be utilized to turn the topological propertieson and off. We discuss various methods to realize such aperturbation using experimentally accessible electromag-netic radiation in the microwave-THz regime.

TOPOLOGICAL TRANSITION IN HgTe/CdTeHETEROSTRUCTURES

Below we outline a proposal for the realization of aFTI in Zincblende structures such as the HgTe/CdTe het-erostructure which are in the trivial phase (d < dc in Ref.5). Consider the Hamiltonian

H(kx, ky) =

(H(k) 0

0 H∗(−k)

), (4)

where

H(k) = ε(k)I + d(k) · σ, (5)

k = (kx, ky) is the two dimensional wave vector, andσ = (σx, σy, σz) are the Pauli matrices. The vector, d(k),is an effective spin-orbit field. Equations (4) and (5) rep-resent the effective Hamiltonian of HgTe/CdTe hetero-junctions [5]. In these systems, the upper block H(k) isspanned by states with mJ = (1/2, 3/2), while the lowerblock is spanned by states with mJ = (−1/2,−3/2). Thelower block H∗(−k) is the time reversal partner of theupper block, and in the following we shall focus our at-tention to the upper block of Eq. (4) only.

The Hamiltonian (5) is diagonalized via a k-dependent SU(2) rotation, that sets the z-axis for thepseudo-spin along the vector d(k). There are two double-degenerate bands with energies ε±(k) = ε(k) ± |d(k)|.Within each sub-block, the TKNN formula provides thesub-band Chern number [22], which for the Hamilto-nian (5) can be expressed as an integer counting the num-

ber of times the vector d(k) wraps around the unit sphereas k wraps around the entire FBZ. In integral form, it isgiven by

C± = ± 1

4π

∫d2k d(k) ·

[∂kx d(k)× ∂ky d(k)

], (6)

where d(k) = d(k)/ |d(k)| is a unit vector and the (±) in-dices label the two bands.

This elegant mathematical construction yields alsoimportant physical consequences, as it is related to thequantized Hall conductance associated with an energyband,

σxy =e2

hC. (7)

Around the Γ point of the first Brillouin zone (FBZ)we can expand the vector d(k) as [5, 23]

d(k) =(Akx, Aky,M −Bk2

), (8)

where the parameters A < 0, B > 0 and M depend onthe thickness of the quantum well and on parameters ofthe materials. We can easily see that the Chern numberimplied by d(k) depends crucially on the relative sign

3

FIG. 1. Energy dispersion ε(k) and pseudospin configuration

−d(k) for the original bands of H(k) in the non topologicalphase (M/B < 0). The non-topological phase is characterizedby a spin-texture which does not wrap around the unit sphere.Upon application of a periodic modulation of frequency ωbigger than the band gap, a resonance appears; the greencircles and arrow depict the resonance condition.

of M and B. Within the approximation of Eq. (8), faraway from the Γ point, d(k) must point south (in thenegative z direction). At the Γ point, d(k) is point-ing north for M > 0, but south for M < 0. For thesimplified band structure, the Chern numbers are clearlyC± = ± [1 + sign(M/B)] /2. For a generic band struc-ture corresponding to Eq. (8) near the Γ-point, the samelogic applies, and we can easily see that a change of signin M induces a change of the Chern number, C, by 1.

We now show that a similar non-trivial topologicalstructure can be induced in such quantum wells, start-ing with the non-topological phase, via periodic modu-lation of the Hamiltonian, which allows transitions be-tween same-momentum states with energy difference of~ω. This creates a circle in the FBZ where transitions be-tween the valence and conduction band are at resonance(see Fig. 1). We intend to use the modulation to reshufflethe spectrum such that the resulting valence band con-sists of two parts: the original valence band outside theresonance circle drawn in Fig. 1, and the original conduc-tion band inside the resonance circle, near the Γ point.From Fig. 1, we see that this indeed leads to the desiredstructure, with the reshuffled pseudospin configurationpointing south near the Γ point and north at large k-values (for M < 0). On the resonance circle, we expectan avoided crossing separating the reshuffled lower bandfrom the upper band.

FLOQUET TOPOLOGICAL INSULATOR IN ANON-EQUILIBRIUM (Cd,Hg)Te

HETEROSTRUCTURE

Let us next consider the Floquet problem in aZincblende spectrum in detail. We add a time depen-dent field to the Hamiltonian (5)

V (t) = V · σ cos(ωt), (9)

where V is a three-dimensional vector, which has to becarefully chosen to obtain the desired result. It is con-venient to transform the bare Hamiltonian to a “rotat-ing frame of reference” such that the bottom band isshifted by ~ω. This is achieved using the unitary trans-formation U(k, t) = P+(k) + P−(k)eiωt, where P±(k) =12

[I ± d(k) · σ

]are projectors on the of upper and lower

bands of H(k). This results in the following Hamiltonian:

HI(t) = P+(k)ε+(k) + P−(k) [ε−(k) + ω] + U V (t)U†,(10)

where ε±(k) are the energies corresponding to P±(k). Inthe “rotating” picture, the two bands cross if ω is largerthen the gap M . The second term in the right-hand-sideof Eq. (10) is the driving term, which directly couples thebands and has a time-independent component.

The solution of HI can also be given in terms of aspinor pointing along a unit vector, nk, which will playthe same role as d(k) for the stationary H(k). nk willencode the topological properties of the FTI.

HI is solved by the eigenstates |ψ±I (k)〉, which for thevalues of momenta, k, away from the resonance ring areonly weakly modified compared to the equilibrium, V =0, case. We define the vector nk = 〈ψ−I (k)|σ|ψ−I (k)〉,which characterizes the pseudospin configuration in thelower (−) band of HI . The vector nk is plotted in Fig. 2for M/B < 0. Indeed, near the Γ point we see thatnk points towards the south pole, and for larger valuesof k, the band consists of the original lower band, andtherefore nk points towards the northern hemisphere forthese k values. These two regimes are separated by theresonance ring, denoted by γ, for which ω = e+(k) −e−(k) (the green curve in Fig. 1).

The topological aspects of the reshuffled lower banddepend crucially on the properties of nk on γ, which are,in turn, inherited from the geometric properties of thedriving potential, encoded in V. These are best illus-trated by employing the rotating wave approximation,as we shall proceed to do below. An exact numericalsolution will be presented in the next section.

The driving field V (t) contains both counter-rotatingand co-rotating terms. In the rotating wave approxima-tion it is given by

VRWA = P+(k) (V · σ) P−(k) + P−(k) (V · σ) P+(k).(11)

4

FIG. 2. Pseudospin configuration nk (blue arrows) and dis-persion of the lower band of HI . Note the dip in the energysurface near k = 0, resulting from the reshuffling of the lowerand upper bands of H(k).

Next, we decompose the vector V as follows

V =(V · d(k)

)d(k) + V⊥(k). (12)

A substitution in Eq. (11) gives

VRWA = V⊥ · σ. (13)

On the curve γ we have:

nk = −V⊥(k)/|V⊥(k)|. (14)

We can define a topological invariant CF similar to Cin Eq. (6), by replacing d(k) with nk. In order for nk

to have a non-vanishing CF , it needs to wrap aroundthe unit sphere. A necessary condition is that on thecurve γ, it forms a loop that winds around the northpole. Now, V⊥(k) lies on the plane defined by d(k) and

V. For the values of k which are on-resonance, d(k)traces a closed loop on the unit sphere which encirclesthe north pole. Therefore, if the driving field vector Vpoints to a point on the Bloch sphere which is encircledby this loop, V⊥(k) will also trace (a different) loop onthe sphere which encircles the north pole, as illustratedby Fig. 3.

Under the conditions stated above and with M < 0,the vector field nk starts from the south pole at Γ pointand continues smoothly to the northern hemisphere forlarger values of |k| while winding around the equator. For

values of k further away from the curve γ, nk ≈ −d(k) asthe driving field is off resonance there. The contributionof these k’s to CF is therefore equal to their contributionto C. Therefore it is evident that CF± = C± ± 1. Notethat for M > 0, CF± = C± ∓ 1.

A comment is in order regarding the time depen-dence of CF . Since the solutions to the time dependentSchrodinger equation are given by transforming back tothe Schrodinger picture,

|ψ±(t,k)〉 = U(t)|ψ±I (k)〉, (15)

FIG. 3. The geometrical condition for creating topologicalquasi-energy bands. The green arrow and circle depicts d(k)on the curve γ in the FBZ for which the resonant conditionholds. The red arrow and cone depicts V⊥(k) on γ. Theblue arrow depicts the driving field vector V. As long asV points within the loop traced by d(k), the vector V⊥(k)winds around the north pole, which is indicated by the blackarrow.

the pseudospin configuration in the Brillouin zone ofthese solutions ,

nk(t) = 〈ψ−(k, t)|σ|ψ−(k, t)〉, (16)

will also depend on time. As long as HI is non-degeneratein the FBZ, which implies that nk(t) is well defined, CF

will be quantized to an integer. As for C, also CF is atopological invariant which is robust to smooth changesin nk(t) which are not singular at any point in the FBZ.Therefore, CF does not depend on time, although thepseduospin configurations nk(t) do, and we can calculateCF using nk.

NON-EQUILIBRIUM EDGE STATES

One of the most striking results of the above consider-ations is the existence of helical edge states once the timedependent field is turned on. Below we demonstrate theformation of edge states in a tight binding model whichcontains the essential features of Eq. (5). The Fouriertransform of the spin-orbit coupling vector, d(k) in thecorresponding lattice model is given by, c.f., Eq. (8),

d(k) = (A sin kx, A sin ky, M − 4B + 2B[cos kx + cos ky]) .

(17)

We consider the above model with the time dependentfield of the form V0σz cos(ωt) in the strip geometry, withperiodic boundary condition in the x direction, and van-ishing boundary conditions at y = 0, L.

We solve the Floquet equation numerically by movingto frequency space and truncating number of harmonics.The wave vector kx is therefore a good quantum number,and the solutions Φ(t) are characterized by ε and kx. Thequasi-energies for this geometry is displayed in Fig. 4.

5

0

0.5

1

1.5

2

0

-1.5

0

1.5

ε

π/2

kx

π0-π-π

-π π

FIG. 4. Quasi-energy spectrum of the Floquet equation (3) ofthe Hamiltonian (17), in the strip geometry: periodic bound-ary conditions in the x direction, and vanishing ones in they direction. The driving field was taken to be in the z di-rection. The horizontal axis labels the momentum kx. Thevertical axis labels the quasi-energies in units of |M |. Twolinearly dispersing chiral edge modes traverse the gap in thequasi-energy spectrum. The parameters used are ω = 2.3|M |,|V| = A = |B| = 0.2|M |. The inset shows the dispersion ofthe original Hamiltonian (17), for the same parameters.

The quasi-energies of the bottom and top band representmodes which are extended spatially, while for each valueof kx there are two modes which are localized in the ydirection.

As is evident from Fig. 4, the quasi-energies of thesemodes disperse linearly, ε(kx) ∝ kx, hence they arepropagating with a fixed velocity. Consider a wavepacket which is initially described by f0(kx). Fromequation (3) we see that it will evolve into ψ(t) =∫dkxe

iε(kx)tf0(kx)Φekx(y, t), where Φekx denotes the quasienergy edge states with momentum kx. Clearly, this willgive a velocity of 〈x〉 =

∫dkx|f(kx)|2 ∂ε

∂kx.

In general, the solutions Φε,kx(t) are time-dependent.An important finding is that the density edge modes areonly very weakly dependent on time. This can be seenin Fig. 5, in which we plot the time dependence of thedensity profile of these modes.

EXPERIMENTAL REALIZATION OF THE FTI

To experimentally realize the proposed state, weneed identify a proper time-dependent interaction in theHgTe/CdTe wells. Below we consider several options,of which the most promising one uses a periodic electric

FIG. 5. Density of edge mode as function of time, |φ(y, t)|2,(a) for kx = 0, and (b) for kx = 0.84, where the edge modesmeet the bulk states. The horizontal axis display the distancefrom the edge, y, in units of the lattice constant, and the timein units of 2π/ω. Only the density for the 20 lattice sitesclosest to the edge are shown for clarity.

field, and the strong linear Stark effect that arises due tothe unique band structure.

Magnetic field realization – Perhaps the simplest real-ization of a time dependent perturbation of the form (9)is by a microwave-THz oscillating magnetic field, polar-ized in the z direction. The effect of Zeeman energiesin thin Hg/CdTe quantum wells can be evaluated by re-calling that the effective model (4) includes states withmJ = ±(1/2, 3/2) in the upper and lower block respec-tively. This would result in an effective Zeeman splittingbetween the two states in each block [23]. The valuefor the g-factor for HgTe semiconductor quantum wellswas measured to be g ≈ 20 [24]. Therefore, a gap inthe quasienergy spectrum on the order of 0.1K can beachieved using magnetic fields of 10mT. Bigger gaps maybe achieved by using Se instead of Te, as its g-factor isroughly twice as large [25].

As can be seen by inspecting Eq. (12), the Chernnumbers CF for each block in this realization depend onlyon the winding of the vector d(k). Therefore, the twoblocks will exhibit opposite CF , resulting in two counter -propagating helical edge modes. As we explain in thenext section, the counter-propagating edge modes cannotcouple to open a gap in the quasi-energy spectrum, eventhough a magnetic field is odd under time reversal.

Stress Modulation – Stress modulation of the quan-tum wells using piezo-electric materials, would lead tomodulation of the parameter M in (5), and leads to twocounter propagating edge states.

Electric field realization – An in-plane electric field isperhaps the most promising route to the FTI, can pro-duce large gaps in the quasi-energy spectrum (comparedthe Zeeman case), and leads to robust co-propagatingedge modes. The electric field is given by

E = Re(E · exp iωt)i∇k. (18)

6

Inserting this into Eq. (11), we get

V⊥(k) = d(k)× (ReE ·∇k)d(k)− (ImE ·∇k)d(k). (19)

As before, the vector field V⊥(k) is orthogonal to d(k),and again, we would like it to wind around the northpole. Now if we take E = E(−ix + y) we get, expandingEq.(19) to second order in kx, ky,

V⊥(k) =A(A2 − 4BM)E

M3

[12 (k2x − k2y)x + kxkyy

]. (20)

Evidently, the vector field V⊥(k) winds twice around theequator. Therefore, for the above choice of E, the Chernnumbers will be CF± = ±2. We note that for the lowerblock will have CF = 0. Therefore, each edge of thesystem will have two co-propagating chiral modes, withspin in the sector of mJ = 1/2, 3/2. Naturally, a choiceof E = E(ix + y) will give CF± = ∓2 for the lower blockand CF = 0 for the upper block. The evolution with theoscillating electric field is topologically distinct from thetrivial one, as the two co-propagating edge modes can-not become gapped. For HgTe/CdTe quantum wells withthickness of 58A[5], we have |V(k)/E| ≈ 1mm , whichleads to a gap in the quasi-energy spectrum on the or-der of 10K already for modest electric fields on the orderof 1V

m which are experimentally accessible with powers< 1mW. We note that decreasing the well thickness in-creases [26] the value of |A/M |, which can help achieveeven larger gaps in the quasi energy spectrum.

DISCUSSION

In summary, we showed that the quasi-energy spec-trum of an otherwise ordinary band insulator irradi-ated by electromagnetic fields can exhibit non-trivialtopological invariants and chiral edge modes. A real-ization of these ideas in Zincblende systems, such asHgTe/CdTe semiconducting quantum wells, can lead toFloquet Topological Insulators that support either co- orcounter-propagating helical edge modes. The Floquet op-erators of these realizations belong, respectively, to sym-metry classes analogous to classes A (no symmetry) andAII (time reversal symmetry with T 2 = −1) in [20].

The symmetry class of the Floquet Topological In-sulator indeed requires careful consideration when twocounter-propagating edge states are present, as in the os-cillating magnetic-field realization suggested in the pre-vious section. In time independent systems, topologicalphases exhibiting counter-propagating edges are only dis-tinct from trivial phases under the restriction T HT −1 =H, where T is the anti-unitary time reversal operator sat-isfying T 2 = −1. In the time-periodic case, the Hamil-tonian at any given time may not possess any symmetryunder time reversal. Nevertheless, when the condition

T H(t)T −1 = H(−t+ τ) (21)

holds (for some τ), the Floquet matrix of Eq. (3) satis-fies T HF (k)T −1 = HF (−k), where T is an anti-unitaryoperator which is related to T by T = V †T V , with V =Sk (−(T + τ)/2), c.f. Eq. (3). Clearly, T 2 = −1. There-fore, under this condition, the quasi-energy spectrumconsists of analogues to Kramer’s doublets, which cannotbe coupled by the Floquet matrix. The counter propagat-ing edge-modes are such a Kramer’s pair, which, there-fore, cannot couple and open a gap (in the quasi-energyspectrum) under any perturbations satisfying Eq. (21).We note that Eq. (21) holds for any Hamiltonian of theform H(t) = H0+V cos(ωt+φ), with time reversal invari-ant H0, and V having unique parity under time reversal,i.e., T V T −1 = ±V . An oscillating magnetic field, beingodd under time reversal, therefore obeys Eq. (21) andleads to two counter propagating edge modes.

Another important question that we did not touchupon is the non-equilibrium onset and steady states [27]of the driven systems. We emphasize that in the presenceof the time dependent fields, response functions includ-ing Hall conductivity will be determined not only by thespectrum of the Floquet operator, but also by the dis-tribution of electrons on this spectrum. These in turndepend on the specific relaxation mechanisms present inthe system, such as electron-phonon mechanisms [28, 29]and electron-electron interaction [30]. An interesting ob-servation here is that the average energies of the edgemodes lie in the original bandgap.

One way to minimize the unwanted non-equilibriumheating effects would be to use an adiabatic build-up ofthe Floquet topological insulator state, e.g. with the fre-quency of the modulation gradually increasing from zero.This should result, at least initially, in a filled lower Flo-quet band. Nevertheless, the scattering events will leadto a relaxation from the conduction band into the va-lence band of the original system and will always playrole by producing mobile bulk quasi-particles. Perhapsthis relaxation could be suppressed by restricting the cor-responding optical modes in the environment. An anal-ysis of the non-equilibrium states of the system will bethe subject of future work.

We thank Joseph Avron, Assa Auerbach, Erez Berg,Andrei Bernevig, James Eisenstein, Lukasz Fidkowski,Victor Gurarie, Israel Klich, and Anatoli Polkovnikov forilluminating conversations. This research was supportedby DARPA (GR, VG), NSF grants PHY-0456720 andPHY-0803371 (GR, NL). NL acknowledges the financialsupport of the Rothschild Foundation and the Gordonand Betty Moore Foundation.

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